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By Richard A. Hudson

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J'^p) with the understanding that (vi-i^Vi) G A holds for all i G { 1 , . . ,p}. The length of this path is p. 2 Shortest Path'Problems with Resource Constraints 2,1 37 Resource feasible p a t h s The description of feasible paths provides a basis for the generic definition of the SPPRC. t. t. path-structural constraints. This section focuses on the first aspect while path-structural constraints are discussed in the next section. , the travel times tij and time windows [a^, hi] in the SPPTW).

6 lists the hues (in general, hyperplanes) defined by p G P , with an intercept of Cp and a slope of 14 — tp. 4. Observe that for ix given, the line of smallest cost defines the value of function I/(7r). The Lagrangian function L(7r) is therefore the lower envelope of all lines and its topmost point corresponds to the value C of the Lagrangian dual problem. If one starts at TT — 0, the largest possible value is L(0) = 3, on the fine defined by path 1246. At that point the slope is negative (the fine 1 A Primer in Column Generation 23 is defined by 3 — 47r) so that the next multipher should be found on the left to the current point.

2000). On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Operations Research,, 48(1):111-128. Vanderbeck, F. (2004). Implementing mixed integer column generation. In this book. Vanderbeck, F. A. (1996), An exact algorithm for IP column generation. Operations Research Letters^ 19:151-159. , and Soumis, F. (2003). On compact formulations for integer programs solved by column generation. Les Cahiers du GERAD G-2003-06, HEC, Montreal, Canada.

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A grammatical study of Beja by Richard A. Hudson

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